Multivariate Random Variables

5. RANDOM VECTORS • The concepts learned for one and two RVs are now extended to multiple RVs • This is important because real-world problems often times depend on many variables (or are said to be multidimensional). • Vectors are convenient for describing multiple RVs as it leads to concise notation and easier math manipulation • The most important random vectors are those that are Gaussian distributed Lecture Notes: Probability and Stochastic Processes 2495.1 Probability Models for N Random Variables • These models just generalize the concepts already seen for 1 and 2 RVs Lecture Notes: Probability and Stochastic Processes 250Lecture Notes: Probability and Stochastic Processes 251Lecture Notes: Probability and Stochastic Processes 2525.2 Vector Notation • Vectors and (nonrandom) matrices will be denoted with a bold type font. E.g. A, X could represent either nonrandom matrices or random vectors. Whether they are random or not should be clear from the context! • A vector sample value (which is not random), uses a lowercase bold letter Lecture Notes: Probability and Stochastic Processes 253• We use similar notation for a function of a random vector: g(X) = g(X1,…,Xn) and a function of n numbers g(x) = g(x1,…,xn) Lecture Notes: Probability and Stochastic Processes 254• Notice that we can concatenate the random vectors X and Y (where X = [X1,…,Xn] and Y = [Y1,…,Yn]) to a random vector W = [X’ Y’] = [X1,…,Xn,Y1,…,Yn] such that the following are equivalent FW(w) = FX,Y(x,y) Lecture Notes: Probability and Stochastic Processes 255Lecture Notes: Probability and Stochastic Processes 256Lecture Notes: Probability and Stochastic Processes 2575.3 Marginal Probability Functions • As in the 2 RV case, we can integrate (or sum) over one or more RVs of a joint PDF and end up with the desired marginal PDFs Lecture Notes: Probability and Stochastic Processes 258Lecture Notes: Probability and Stochastic Processes 259Lecture Notes: Probability and Stochastic Processes 260Lecture Notes: Probability and Stochastic Processes 2615.4 Independence of Random Variables and Random Vectors • Whenever a joint PDF or PMF can be written as the product of its individual 1 RV marginal distributions, these RVs are said to be independent Lecture Notes: Probability and Stochastic Processes 262Lecture Notes: Probability and Stochastic Processes 263Other Definitions Lecture Notes: Probability and Stochastic Processes 264Lecture Notes: Probability and Stochastic Processes 2655.6 Expected Value Vector and Correlation Matrix • For matrices and vectors, statistical operations are performed component-wise! Lecture Notes: Probability and Stochastic Processes 266Vector Correlation and Covariance Lecture Notes: Probability and Stochastic Processes 267Lecture Notes: Probability and Stochastic Processes 268Lecture Notes: Probability and Stochastic Processes 269Lecture Notes: Probability and Stochastic Processes 270Lecture Notes: Probability and Stochastic Processes 271Lecture Notes: Probability and Stochastic Processes 272Lecture Notes: Probability and Stochastic Processes 2735.7 Gaussian Random Vectors • A random vector where all components are Gaussian distributed is said to be jointly Gaussian. It is also known as a Gaussian random vector • The PDF of the Gaussian random vector is given by • The diagonal elements of CX[i,i] = E[XiXi] = E[Xi2] = Var[Xi] = σi2 • The off-diagonal elements are CX[i,j] = E[XiXj] = Cov[Xi,Xj] = σij • A diagonal covariance matrix can be denoted by CX = diag[σ12, σ22,…, σn2] Lecture Notes: Probability and Stochastic Processes 274Lecture Notes: Probability and Stochastic Processes 275Lecture Notes: Probability and Stochastic Processes 276Lecture Notes: Probability and Stochastic Processes 277Lecture Notes: Probability and Stochastic Processes 278Lecture Notes: Probability and Stochastic Processes 279Lecture Notes: Probability and Stochastic Processes 280• A = UDU’ is the eigendecomposition of A (but can also be obtained using the singular value decomposition) • In Matlab: [U,D]=eig(A); Lecture Notes: Probability and Stochastic Processes 281Lecture Notes: Probability and Stochastic Processes 282